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16. Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consist- ing of the players that tied for first place, the next group being those that tied for the next-best posi- tion, and so on. Let N(n) denote the number of dif- ferent possible outcomes. For instance, N(2) = 3, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first. (a) List all the possible outcomes when n = 3. (b) With N(0) defined to equal 1, argue, without any computations, that =(7) NO | N(n − i N(n) = Hint: How many outcomes are there in which i players tie for last place? (c) Show that the formula of part (b) is equivalent to the following: #-1 N(n) = Σ(1) NO i=0 (d) Use the recursion to find N(3) and N(4).

16. Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consist- ing of the players that tied for first place, the next group being those that tied for the next-best posi- tion, and so on. Let N(n) denote the number of dif- ferent possible outcomes. For instance, N(2) = 3, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first. (a) List all the possible outcomes when n = 3. (b) With N(0) defined to equal 1, argue, without any computations, that =(7) NO | N(n − i N(n) = Hint: How many outcomes are there in which i players tie for last place? (c) Show that the formula of part (b) is equivalent to the following: #-1 N(n) = Σ(1) NO i=0 (d) Use the recursion to find N(3) and N(4).

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.6: Counting Principles
Problem 74E: Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is...
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16. Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consist- ing of the players that tied for first place, the next group being those that tied for the next-best posi- tion, and so on. Let N(n) denote the number of dif- ferent possible outcomes. For instance, N(2) = 3, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first. (a) List all the possible outcomes when n = 3. (b) With N(0) defined to equal 1, argue, without any computations, that =(7) NO | N(n − i N(n) = Hint: How many outcomes are there in which i players tie for last place? (c) Show that the formula of part (b) is equivalent to the following: #-1 N(n) = Σ(1) NO i=0 (d) Use the recursion to find N(3) and N(4).
Transcribed Image Text:16. Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consist- ing of the players that tied for first place, the next group being those that tied for the next-best posi- tion, and so on. Let N(n) denote the number of dif- ferent possible outcomes. For instance, N(2) = 3, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first. (a) List all the possible outcomes when n = 3. (b) With N(0) defined to equal 1, argue, without any computations, that =(7) NO | N(n − i N(n) = Hint: How many outcomes are there in which i players tie for last place? (c) Show that the formula of part (b) is equivalent to the following: #-1 N(n) = Σ(1) NO i=0 (d) Use the recursion to find N(3) and N(4).
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